Momentum ji ji Filtered i i i i - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

Outline

• Filtering
• Large Scales and Subgrid Scales
• Subgrid Scales Stresses
• Leonard Stress
• Smogorinski Model
• Cross Stress
• G. Ahmadi

ME 639-Turbulence

Mass

x v

i i 

  x v

i i 

 

i j ji j i j i

f x t x v v t v                   

i j ji j i j i

f x t x v v t v                   

 

i , j j , i ij ij

v v p t      

 

i , j j , i ij ij

v v p t      

Momentum Newtonian

• G. Ahmadi

ME 639-Turbulence

i i i

v v v   

i i i

v v v    p p p    p p p   

ij ij ij

t t t   

ij ij ij

t t t   

     



     

D

d , G x x x x x

     



     

D

d , G x x x x x

         

Large Scale Subgrid Scale Decomposition

Note That

Filter (Gaussian)

Filtered Large Scale

SLIDE 2

2

• G. Ahmadi

ME 639-Turbulence

x v

i i 

  x v

i i 

 

i j S ji j ji j i j i

f x t x t x v v t v                      

i j S ji j ji j i j i

f x t x t x v v t v                      

 

i j i j S ji

v v v v t    

 

i j i j S ji

v v v v t    

Subgrid-Scale Stress Tensor

• G. Ahmadi

ME 639-Turbulence

C ji L ji R ji S ji

t t t t   

C ji L ji R ji S ji

t t t t   

i j R ji

v v t     

i j R ji

v v t     

 

i j i j L ji

v v v v t    

 

i j i j L ji

v v v v t    

 

j i i j C ji

v v v v t      

 

j i i j C ji

v v v v t      

Reynolds Stresses Leonard Stresses Cross Stresses

• G. Ahmadi

ME 639-Turbulence

Smagorinsky Model

ji T R ji

D 2 t  

ji T R ji

D 2 t  

             

j i i j ji

x v x v 2 1 D              

j i i j ji

x v x v 2 1 D

  

2

1 k k 2 S T

D D c

 

  

  

2

1 k k 2 S T

D D c

 

  

 3

1 3 2 1

    

 3

1 3 2 1

    

21 . cS  21 . cS 

Smagorinsky Constant Grid Size

• G. Ahmadi

ME 639-Turbulence

Speziale Galilean Invariance

 

i j i j r c ji

v v v v c t    

 

i j i j r c ji

v v v v c t     1 cr  1 cr 

 

i j i j L ji

v v v v t    

 

i j i j L ji

v v v v t    

Leonard Stresses are Evaluated

SLIDE 3

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• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence (a) (b)

• G. Ahmadi

ME 639-Turbulence

SLIDE 4

4

• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

• LES captures computes for the

large eddies of turbulence while the sub-grid scales are modeled.

• LES can provide more accurate

description of turbulent flows.

• Smagorinski

model provide a reasonable approximation.

• G. Ahmadi

ME 639-Turbulence